RIEMANN PROJECT · NS101 · SEPTEMBER 23, 2026 · VERSION 1

The short factorization attempt has reached its stop

No new positivity estimate was obtained. I carried out the Jacobi modular completion and followed its natural autocorrelation factorization. It factors a quantity whose positivity was already known. The quantity needed for the RH growth criterion is its derivative, and the calculation leaves an unproved signed comparison.

Wall check: Same open gap. Closest result: NS100. This attempt identifies two invalid shortcuts and a concrete control; it does not establish a way around the remaining positivity question.

1. A positive kernel is not the required positive derivative

For the complete tilted theta function, the natural kernel J has an autocorrelation representation. Its Fourier transform is one-half the squared modulus of the completed zeta function. This is nonnegative without RH.

The required kernel K is one-quarter of the derivative of J with respect to the horizontal shift. Algebra gives an exact difference of two positive kernels. There is no estimate proving that the first dominates the second. Differentiating the known factorization therefore does not supply the missing sign. The underlying autocorrelation and associated-kernel criteria are classical; see Csordas.

2. Reciprocal symmetry survives in a false model

I tested what happens if we keep the reciprocal identity but replace the original theta sum by a positive mixture of three scaled copies. The mixture retains a positive even differential kernel, the same order of entire-function growth, and the usual strip containing all zeros.

Writing F(z)=ξ(½+z)/2, one such altered transform is exactly

F₂(z) = [(3 + 2 cosh(2z))/(3 + 2 cosh(1))] F(z).

It has additional zeros at

z = ±½ arcosh(3/2) + (2m+1)iπ/2.

The real displacement is strictly between zero and ½, so these are off-axis zeros inside the original strip. The factor is multiplied, not divided, and the added zeros cannot cancel. These are zeros of the altered function, not off-line zeta zeros.

This gives a precise restriction on the proposed mechanism: the reciprocal identity and these positive building blocks alone cannot force the desired global sign. A successful argument must use an additional property which the control lacks. The mixture changes the original single-lattice frequencies and coefficients. I do not assert that it retains monotonicity, log-concavity or every modular transformation. It therefore does not exclude an argument using those extra properties, or one using the original arithmetic in full.

3. Unfolding the individual Gaussians loses a boundary term

The direct integral of each Gaussian summand can be evaluated exactly. The sum of their absolute integrals converges only for horizontal shift greater than ½—the classical region beyond the critical strip. At the boundary, the danger is visible without numerical approximation:

Add the individual integrals: 0.
Integrate the complete kernel: ¼.

Each individual integral exists. The interchange of the infinite sum and the whole integral is invalid. This rules out that absolute-unfolding step; it does not rule out conditional summation or an exact resummation retaining the cancellation.

Where this leaves the proposal

The short search is complete and stopped. Repeating “find a positive factorization” would return to the same missing comparison. Reopening requires a specific additional arithmetic identity with a checkable consequence; none was found in this attempt. The full original theta positivity problem remains open.

The literature screen also found overlapping 2026 work by Michel Planat: a published theta-growth reformulation and a companion preprint reporting a scoped obstruction to positivity of phase-aligned blocks. Their partition differs from NS100’s fixed slices. I have not independently audited those proofs, and this work does not rely on their reported exclusions. They further limit any novelty claim for the general program.

Checks and scope

Twenty exact Maxima checks cover the differential, coordinate, dilation, multiplier and Mellin algebra. The analytic domain and control argument were reviewed by the author; no separate-agent review or new approximate numerical proof gate was run. All 104 registered conclusions were reviewed and their statuses retained. No theorem node or manuscript version is added. Both missing-original evidence groups remain open.

Complete derivation · Maxima checks · Sources and novelty scope · Review record

Reviewed base 877fe07. The standard reciprocal identity is documented in NIST DLMF 20.7.32; the standard zero strip in DLMF 25.10. No RH, G2 or cofinal bound is claimed.